---
title: q-Analogue of Rational Numbers
url: https://www.emergentmind.com/topics/q-analogue-of-a-rational-number
type: topic
---

# q-Analogue of Rational Numbers

Searching arXiv for recent and foundational papers on q-deformed rationals and q-analogs of rational numbers.
A q-analogue of a rational number, in the sense introduced by Morier-Genoud and Ovsienko, is a canonical rational function in the parameter \(q\) attached to a classical rational \(x\in\mathbb{Q}\cup\{\infty\}\). It is designed so that specialization at \(q=1\) recovers \(x\), while the deformation remains compatible with the modular action of \(PSL(2,\mathbb{Z})\), continued fractions, and Farey geometry. For \(x=n/m\) in lowest terms, one obtains
\[
[x]_q=\frac{N(q)}{M(q)},
\qquad N(q),M(q)\in\mathbb{Z}[q],
\]
with positive coefficients; this deformation is not the naive ratio \([n]_q/[m]_q\), but a genuinely two-variable construction in which numerator and denominator data are intertwined [2503.23834].

## 1. Definition by modular invariance

The modern definition is formulated on the projective line \(\mathbb{Q}\cup\{\infty\}\), where \(PSL(2,\mathbb{Z})\) acts by fractional-linear transformations
\[
\begin{pmatrix}a&b\\ c&d\end{pmatrix}(x)=\frac{ax+b}{cx+d}.
\]
Its \(q\)-deformed counterpart acts on \(\mathbb{Z}(q)\) through
\[
T_q(X)=qX+1,\qquad S_q(X)=-\frac{1}{qX},
\]
and these satisfy the same defining relations as the classical generators:
\[
S_q^2=(T_qS_q)^3=\mathrm{Id}.
\]
The \(q\)-rational associated to \(x\) is the unique map \(R_q(x)=[x]_q\) characterized by \(PSL(2,\mathbb{Z})\)-equivariance,
\[
[A(x)]_q=A_q([x]_q),
\]
together with the normalization \([0]_q=0\) [2503.23834].

Equivalent axiomatic recurrences are
\[
[x+1]_q=q[x]_q+1,\qquad [-1/x]_q=-\frac{1}{q[x]_q},
\]
starting from \([0]_q=0\). For rationals, these determine \([x]_q\) uniquely and imply the auxiliary identities
\[
[-x]_q=-q^{-1}[x]_{q^{-1}},\qquad [1/x]_q=\frac{1}{[x]_{q^{-1}}},
\]
which make the modular character of the theory explicit [2102.00891].

The specialization property is fundamental:
\[
\lim_{q\to 1}[x]_q=x.
\]
For \(x=n/m\) in lowest terms, the corresponding polynomials satisfy \(N(1)=n\) and \(M(1)=m\) [2503.23834].

## 2. Continued fractions, matrices, and explicit formulas

Every rational admits two compatible continued-fraction descriptions, and each leads to a closed \(q\)-formula. For the Hirzebruch, or minus-sign, continued fraction
\[
\frac{n}{m}=\llbracket c_0,c_1,\dots,c_\ell\rrbracket,
\]
the deformation is obtained by replacing integers with Gaussian \(q\)-integers and inserting powers of \(q\):
\[
\left[\frac{n}{m}\right]_q
=
[c_0]_q-\cfrac{q^{c_0-1}}{[c_1]_q-\cfrac{q^{c_1-1}}{\ddots-\cfrac{q^{c_{\ell-1}-1}}{[c_\ell]_q}}}.
\]
For the regular continued fraction
\[
\frac{n}{m}=[a_0;a_1,\dots,a_k],
\]
one obtains an equivalent alternating expression involving both \(q\) and \(q^{-1}\) [2503.23834].

These constructions admit a convergent recursion. If
\[
\left[\frac{n_i}{m_i}\right]_q=\frac{N_i(q)}{M_i(q)},
\qquad
\frac{n_i}{m_i}=\llbracket c_0,\dots,c_i\rrbracket,
\]
then
\[
\begin{aligned}
N_{i+1}(q)&=[c_{i+1}]_q\,N_i(q)-q^{c_i-1}N_{i-1}(q),\\
M_{i+1}(q)&=[c_{i+1}]_q\,M_i(q)-q^{c_i-1}M_{i-1}(q).
\end{aligned}
\]
The same rational function is also obtained from \(2\times 2\) \(q\)-matrices such as
\[
T_q=\begin{pmatrix}q&1\\ 0&1\end{pmatrix},
\qquad
S_q=\begin{pmatrix}0&-1\\ q&0\end{pmatrix},
\]
or, in an equivalent presentation,
\[
R_q=\begin{pmatrix}q&1\\ 0&1\end{pmatrix},
\qquad
L_q=\begin{pmatrix}q&0\\ q&1\end{pmatrix},
\]
which encode the continued-fraction word directly [2503.23834; 2511.11290].

Typical values are as follows.

| \(x\) | Continued fraction | \([x]_q\) |
|---|---|---|
| \(1/2\) | \([0;2]\) | \(\dfrac{q}{1+q}\) |
| \(2/3\) | \(\llbracket 1,3\rrbracket\) | \(\dfrac{q+q^2}{1+q+q^2}\) |
| \(3/5\) | \([0;1,1,2]\) | \(\dfrac{q^3+q^2+q}{q^3+q^2+2q+1}\) |
| \(5/3\) | \(\llbracket 2,3\rrbracket\) | \(\dfrac{1+q+2q^2+q^3}{1+q+q^2}\) |

These examples illustrate the general phenomenon that both numerator and denominator have nonnegative coefficients and that evaluation at \(q=1\) reproduces the original fraction [2503.23834].

## 3. Positivity, total positivity, and symmetry

A central structural feature is positivity. For every rational \(x=n/m\), the polynomials \(N(q)\) and \(M(q)\) in
\[
[x]_q=\frac{N(q)}{M(q)}
\]
have positive integer coefficients [2503.23834]. Earlier formulations established the same positivity through continued-fraction recurrences, triangulations, and quiver models [1812.00170].

A stronger statement is total positivity. If \(x>x'\) and
\[
[x]_q=\frac{N(q)}{M(q)},
\qquad
[x']_q=\frac{N'(q)}{M'(q)},
\]
then
\[
\mathcal{X}_{x,x'}(q)=N(q)M'(q)-M(q)N'(q)
\]
has positive integer coefficients. Moreover,
\[
\mathcal{X}_{x,x'}(q)=q^d
\quad\text{if and only if}\quad
nm'-mn'=1,
\]
so Farey-neighbor relations are detected exactly by monomials [2503.23834].

For successive convergents one has the determinant identity
\[
R_n(q)S_{n-1}(q)-S_n(q)R_{n-1}(q)=q^{a_1+\cdots+a_{n-1}},
\]
the \(q\)-analogue of the classical continuant Wronskian formula [1908.04365]. Matrix traces exhibit an additional symmetry: for any \(A\in PSL(2,\mathbb{Z})\), the trace of \(A_q\) is palindromic [2503.23834].

Unimodality is another prominent property. For every rational \(x\), the coefficient sequences of \(N(q)\) and \(M(q)\) are unimodal, extending the classical Sylvester–Cayley phenomenon for Gaussian \(q\)-binomials [2503.23834]. A later combinatorial approach also proves unimodality through rank polynomials of fence posets [2511.11290].

## 4. Farey, snake graphs, posets, quivers, and Schubert geometry

The theory has several equivalent combinatorial realizations. On the Farey graph, the ordinary mediant operation is replaced by a \(q\)-weighted mediant
\[
\frac{N}{M}\oplus\frac{R}{S}
=
\frac{N(q)+q^dR(q)}{M(q)+q^dS(q)},
\]
with the exponent determined inductively from the initial triangle \(\frac01,\frac11,\frac10\). This recursion reproduces exactly the \(q\)-rationals across the Farey tessellation and is the analogue of the weighted Pascal identity for Gaussian \(q\)-binomial coefficients [2503.23834].

In the original construction, a rational \(p/q>1\) is encoded by a triangulation of a polygon. The coefficients of the numerator and denominator then count closure subsets of oriented \(A\)-type quivers dual to that triangulation; equivalently, they count subrepresentations of the maximal indecomposable representation of the corresponding quiver [1812.00170].

A second family of models uses snake graphs. If \(x=[a_1,\dots,a_{2\ell}]\ge 1\), its snake graph is a zig-zag polyomino with \(a_1+\cdots+a_{2\ell}-1\) unit squares. The coefficient of \(q^k\) in the numerator polynomial \(N(q)\) equals the number of NE-lattice paths in the snake graph with \(k\) boxes below. The denominator has an analogous interpretation in a smaller snake graph [2503.23834]. The same numerator polynomial also appears as a perfect-matching generating function, via the specialization of cluster-algebra \(F\)-polynomials [2111.07912].

Fence posets provide an equivalent order-theoretic description. For an even-length regular continued fraction, one constructs a fence poset whose lattice of lower ideals has rank-generating function equal to \(N_{a/b}(q)\) [2111.07912]. A later refinement treats all positive rationals, including those in \((0,1)\), by using a single fence poset or a single snake graph together with an internal partition that simultaneously yields numerator and denominator generating functions [2511.11290].

The geometric interpretation is formulated over finite fields. For a snake graph identified with a skew Young diagram \(\lambda/\mu\), one has
\[
q^{|\mu|}N_{a/b}(q)
=
\left|
\bigcup_{\mu\le \nu\le \lambda}\Omega^\circ_\nu(\mathbb{F}_q)
\right|,
\]
where the union ranges over open Schubert cells in a Grassmannian \(\mathrm{Gr}(k,n)\). Thus the numerator counts \(\mathbb{F}_q\)-points in a union of open Schubert cells, paralleling the classical finite-field interpretation of Gaussian \(q\)-binomials [2111.07912].

## 5. Extension to irrationals and analytic theory

The rational theory extends to real numbers by stabilization. If \(x\ge 1\) is irrational and \(x_n\to x\) is any rational sequence, then the Taylor expansions
\[
[x_n]_q=\sum_{k\ge 0}\varkappa_{n,k}q^k
\]
stabilize coefficientwise: for each fixed \(k\), the coefficient \(\varkappa_{n,k}\) is eventually constant. The limit defines a formal power series
\[
[x]_q=\lim_{n\to\infty}[x_n]_q\in\mathbb{Z}[[q]],
\]
independent of the approximating sequence [2503.23834].

If
\[
x=[a_0;a_1,a_2,\dots]
\]
is the infinite regular continued fraction of an irrational, then \([x]_q\) is given by the corresponding infinite \(q\)-continued fraction, and truncation after \(n\) terms stabilizes at least the first \(n\) Taylor coefficients [2503.23834]. For quadratic irrationals, \([x]_q\) often has a closed radical form
\[
[x]_q=\frac{P(q)+\sqrt{Q(q)}}{R(q)},
\]
with \(Q(q)\) palindromic. For the golden ratio,
\[
\left[\frac{1+\sqrt5}{2}\right]_q
=
\frac{q^2+q-1+\sqrt{(q^2+3q+1)(q^2-q+1)}}{2q}
\]
[2503.23834].

The analytic question is the radius of convergence \(R(x)\) when \(q\) is complex. A universal lower bound holds:
\[
R(x)>3-2\sqrt2\approx 0.1716.
\]
A conjecturally optimal bound is
\[
R(x)\ge \frac{3-\sqrt5}{2}\approx 0.3820,
\]
with equality expected precisely for numbers \(PSL(2,\mathbb{Z})\)-equivalent to the golden ratio; this is the proposed \(q\)-analogue of Hurwitz’s irrational number theorem [2102.00891]. For rational classes whose Hirzebruch–Jung coefficients satisfy \(c_i\ge 4\) eventually, the stronger inequality \(R(x)> (3-\sqrt5)/2\) is proved [2102.00891].

At rational points, stabilization depends on the side of approximation. Besides the “right” value \([x]_q\), there is a “left” value
\[
[x]_q^\flat
=
A_q\!\left(\frac{q-1}{q}\right),
\qquad A(0)=x,
\]
which is independent of the chosen \(A\in PSL(2,\mathbb{Z})\). Approximations from below stabilize to \([x]_q\), while approximations from above stabilize to \([x]_q^\flat\). The two are intertwined by an involution \(\mathcal{I}\) [2503.23834].

## 6. Relations to knot theory, representation theory, and \(q\)-calculus

The construction is part of the broader theory of \(q\)-deformation. Its modular \(q\)-action places it near \(q\)-calculus and the representation-theoretic context in which \(q\)-deformations underlie quantum groups [2503.23834]. The basic recursion
\[
[n+1]_q=q[n]_q+1
\]
is the simplest manifestation of this connection.

For rational knots, or \(2\)-bridge knots in Conway’s parametrization, the normalized Jones polynomial is expressed directly in terms of the numerator and denominator polynomials of the corresponding \(q\)-rational. In one formulation,
\[
J_x(q)=q\,R(x;q)+(1-q)\,S(x;q),
\]
so the knot invariant is a linear combination of the two constituents of \([x]_q\) [1812.00170].

Representation theory and Schubert calculus enter through the finite-field Grassmannian interpretation of the numerator, while cluster algebras enter through snake graphs, perfect matchings, and \(F\)-polynomials [2111.07912]. Discrete integrable systems also appear: in quadratic cases, the coefficients of \([x]_q\) give rise to Somos-type recurrences for associated Hankel determinants [2503.23834].

A distinct arithmetic connection is provided by the \(q\)-analog of Markoff numbers. Using Christoffel words and a matrix morphism \(\mu_q\), one obtains a perfect-matching generating formula whose specialization at \(q=1\) recovers the classical Markoff number [2511.11290].

## 7. Comparisons, misconceptions, and open problems

A persistent misconception is to identify the \(q\)-analogue of \(n/m\) with the naive quotient \([n]_q/[m]_q\). That quotient does recover \(n/m\) at \(q=1\), but it does not satisfy the modular invariance, does not encode the continued-fraction geometry, and has no refined combinatorial or finite-field interpretation. By contrast, the Morier-Genoud–Ovsienko deformation is \(PSL(2,\mathbb{Z})\)-equivariant, positive, unimodal, and compatible with Farey and Schubert structures [2503.23834].

Several questions remain open. The main analytic problem is the \(q\)-Hurwitz conjecture asserting that
\[
R(x)\ge \frac{3-\sqrt5}{2}
\]
for all real \(x>0\), with equality exactly on the \(PSL(2,\mathbb{Z})\)-orbit of the golden ratio [2102.00891]. Other open directions include sharper lower bounds for rational radii of convergence, the characterization of which integer-coefficient power series arise as \(q\)-deformed reals, the algebraicity of radii for general quadratic irrationals, and finer zero-distribution results for Fibonacci and Pell polynomial families associated with convergents [2102.00891].

On the combinatorial side, recent work shows that a single fence poset or a single snake graph can generate both numerator and denominator for every positive rational, extending earlier interpretations that were restricted to rationals \(>1\) or required separate objects [2511.11290]. This suggests that the topic is still being reorganized around a more unified combinatorial infrastructure, while the modular and analytic aspects continue to develop in parallel.

Source: https://www.emergentmind.com/topics/q-analogue-of-a-rational-number